Distributed Remote Estimation Over the Collision Channel With and Without Local Communication

Abstract

Internet of Things networks are the large-scale distributed systems consisting of a massive number of simple devices communicating, typically, over a shared wireless medium. This new paradigm requires novel ways of coordinating access to limited communication resources without introducing unreasonable delays. Herein, the optimal design of a remote estimation system with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> sensors communicating with a fusion center via a collision channel of limited capacity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k\leq n$</tex-math></inline-formula> is considered. In particular, for independent and identically distributed observations with a symmetric probability density function, we show that the problem of minimizing the mean-squared error with respect to a threshold strategy is quasi-convex. When coordination among sensors via a local communication network is available, the online learning of possibly unknown parameters of the probabilistic model is possible, enabling each sensor to optimize its own threshold autonomously. We propose two strategies for remote estimation with local communication: 1) one strategy swiftly reaches the performance of the optimal decentralized threshold policy and 2) the second strategy approaches the performance of the optimal centralized scheme with a slower convergence rate. A hybrid scheme that combines the best of both approaches is proposed, offering fast convergence and excellent performance.